Point defects in two-dimensional colloidal crystals: simulation vs. elasticity theory.

Transcription

1 Point defects in two-dimensional colloidal cystals: simulation vs. elasticity theoy. Wolfgang Lechne Univesity of Vienna, Faculty of Physics, Boltzmanngasse 5, Vienna, Austia. Chistoph Dellago Univesity of Vienna, Faculty of Physics, Boltzmanngasse 5, Vienna, Austia. Received XXXXth Month, 200X Accepted XXXXth Month, 200X DOI: / 1 Intoduction Gaphical abstact: Displacement field (left) and paticle configuation (ight) of an intestitial defect in a two-dimensional colloidal cystal. Using numeical and analytical calculations we study the stuctue of vacancies and intestitials in twodimensional colloidal cystals. In paticula, we compae the displacement fields of the defect obtained numeically with the pedictions of continuum elasticity theoy fo a simple defect model. In such a compaison it is of cucial impotance to employ coesponding bounday conditions both in the paticle and in the continuum calculations. Hee, we fomulate the continuum poblem in a way that makes it analogous to the electostatics poblem of finding the potential of a point chage in peiodic bounday conditions. The continuum calculations can then be caied out using the technique of Ewald summation. Fo intestitials, the displacement fields pedicted by elasticity theoy ae accuate at lage distances, but lage deviations occu nea the defect fo distances of up to 10 lattice spacings. Fo vacancies, the elasticity theoy pedictions obtained fo the simple model do not epoduce the numeical esults even fa away fom the defect. Many popeties of cystalline mateials ae stongly affected by the pesence of impefections in the cystal lattice. In paticula, point defects such as vacancies and self intestitials have a pofound influence on the mechanical, optical, and electical behavio of the mateial. Recent advances in expeimental techniques fo the manipulation and obsevation of colloidal systems [1, 2] now pemit to study the fundamental popeties of point defects in condensed matte systems with atomistic space and time esolution. Using optical tweezes to manipulate individual colloidal paticles, Petsinidis and Ling [3, 4, 5] have geneated point defects in twodimensional cystals and have studied thei stable stuctues, inteactions and diffusion. In othe expeimental wok, Maet, Günbeg and collaboatos [6, 7, 8] have investigated the effective inteactions of themally excited topological defects in cystals of paamagnetic colloidal paticles and discussed the significance of these inteactions fo 2d-melting, which accoding to the celebated Kostelitz-Thouless-Halpein-Nelson-Young theoy [9], involves the fomation and dissociation of topological defect pais. Point defects also play an impotant ole in the two-dimensional electon lattice, the so called Wigne cystal [10], in which they cay implication fo the melting mechanism [11, 12, 13], and fo the the conjectued supesolid phase of Helium 4 [14], in which case the attactive inteactions of vacancies and intestitial may lead to expulsion of defects fom the cystal thus peventing fomation of a supesolid [15]. Fom expeiments [3, 4, 5] and compute simulations [16, 17] it is known that vacancies and intestitials in 2d colloidal cystals can occu in vaious stable configuations with symmeties that diffe fom the symmety of the undelying lattice. In the pesent aticle we study the stuctue and enegetics of such point defects in a 2d cystal of soft sphees using compute simulations and analytical calculations. In paticula, we addess the 1

2 question of how accuately the distubances ceated by point defects can be ationalized in tems of elastic continuum theoy. Due to the long ange natue of elastic displacement fields, in caying out such a compaison it is citical to use coesponding bounday conditions in the paticle and continuum calculation. Simila peiodic image effects due to elastic inteactions need to be taken into account also in the atomistic modeling of dislocations [18, 19]. As we show below, the stuctue of point defects in a system with peiodic bounday conditions can be detemined within elasticity theoy with the technique of Ewald summation familia fom the compute simulation of systems with electostatic inteactions [20]. This technique has been used befoe to adapt the inteaction of dislocations to peiodic bounday conditions [10, 21, 22]. Hee we use Ewald summation to solve the equilibium condition of elasticity theoy and calculate the displacement field of a simple point defect model unde peiodic bounday conditions. While elasticity theoy accuately descibes the lattice distotion caused by point defects in the fa field, non-lineaities and discete lattice effects dominate the defect stuctue nea the defect. Although all the numeical studies discussed in this pape ae caied out fo two-dimensional cystals of soft sphees, simulations pefomed fo thee-dimensional cystals of vaious stuctues and with diffeent inteaction potentials, including Gaussian coe, Lennad-Jones, sceened electostatic, and 1/ 3 -inteactions, indicate that the phenomena descibed hee ae common to many atomic and colloidal systems. The emainde of the pape is oganized as follows. In Sec. 2 we descibe how we detemine the displacement fields of point defects numeically, discuss whethe such calculations should be done at constant pessue o at constant volume, and pesent the displacement fields caused by intestitials and vacancies in vaious configuations. The elasticity theoy fomalism we use to analyze the displacement pattens of point defects is developed in Sec. 3. In this section, we also discuss the analogy between elasticity theoy and electostatics that enables us to use the method of Ewald summation to obtain displacement fields fom elasticity theoy. These displacement fields ae compaed to those obtained numeically in Sec. 4. A summay and conclusions ae povided in Sec Displacement fields Thoughout this pape, we use the Gaussian coe model as a geneic model fo a system of soft sphees [26, 27, 28]. In this puely epulsive system, pais of paticles inteact via v() = ε exp( 2 /σ 2 ) (1) whee is the inte-paticle distance and ε and σ set the enegy and length scales, espectively. In the following, enegies ae measued in units of ε and distances in units of σ. The Gaussian coe model, often studied in soft condensed matte science, accuately descibes the shot-anged effective inteactions between polyme coils in solution [29]. Depending on tempeatue and density, the thee-dimensional Gaussian coe model can exist as a fluid, a bcc- o an fcc-solid [27]. In two dimensions, the pefect tiangula lattice is the lowest enegy stuctue at all densities [30]. Since Gaussian coe paticles ae puely epulsive, they can fom stable cystals only at pessues lage than zeo. The two-dimensional Gaussian coe model, which appoaches the had disk system in the limit of low tempeatue and low density [30], has been used peviously to study the melting tansition in two dimensions [30, 31]. To make contact between numeical calculations in the paticle system and continuum elasticity theoy, we detemine, at T = 0, the displacement field [32] u( i ) i i (2) caused by the intoduction of the defect into the system. Hee, i and i denote the position of paticle i with and without the defect, espectively. The displacemet field completely descibes the esponse of the system s stuctue to the petubation intoduced by the defect. Numeically, we detemine displacement fields by inseting a paticle into o emoving it fom a pefect cystal on a tiangula lattice. The system is then elaxed to a new minimum enegy configuation by steepest descent minimization at constant volume of the simulation box. Peiodic bounday conditions apply. Typically, about tens of thousands of steepest descent steps ae equied to detemine minimum enegy stuctues accuately. In each minimization step, each paticle is moved in the diection of the foce acting on the paticle whee the absolute value of the displacement in chosen to be small enough to ensue that the enegy of the system is educed in each step. The displacement u( i ) of paticle i is then simply the vecto which connects the position of paticle i befoe the minimization, i, with its position afte the minimization, i. The lagest system we study hee consists of N = 199, 680 Gaussian coe paticles (without the exta paticle) at a numbe density of ρ = 0.6σ 2 coesponding to a lattice constant of a = σ. The almost squae simulation box has length L x = 416a and height L y = ( 3/2)480a = a with aspect atio L y /L x = Constant V o constant p? In calculating the displacement fields caused by point defects the question natually aise whethe one should do that at constant volume V o at constant pessue p. Natually, the choice should depend on the paticula expeimental situation one is inteested in. As we will show hee, howeve, the displacement fields caused by a point defect at constant pessue and at constant volume ae simply elated. To detemine how they ae elated, conside a pefect tiangula cystal at T = 0 enclosed in a ectangula cell of volume V 0 with appopiate aspect 2

3 atio. Fo this paticula volume, the cystal is unde the hydostatic pessue p 0. Insetion of a point defect into the cystal at a fixed total volume distots the cystal and atom i is displaced by u V0 ( i ) = i (V 0) i (V 0 ), (3) whee the subscipt V 0 indicates that the displacement field u V0 ( i ) is obtained at constant volume V 0. In the above equation, i (V 0) and i (V 0 ) ae the positions of atom i in the system of volume V 0 with and without the defect, espectively. If one equies, howeve, that the defects is ceated at constant pessue p 0, the volume of the simulation cell changes fom V 0 to V 1 (typically, it will incease fo an intestitial and decease fo a vacancy) and the atoms ae displaced by a diffeent amount, u p0 ( i ) = i (V 1) i (V 0 ), (4) whee the subscipt p 0 implies that the displacement field is consideed at constant pessue. Note that hee we assume that duing the geneation of the defect the simulations cell only expands o contacts, but does not change its shape. This assumption can be lifted as discussed below. We now imagine that the defect geneation at constant pessue is caied out in two steps: fist the system is homogeneously dilated without defect fom volume V 0 to volume V 1 ; in the second step, the defect is inseted at constant volume V 1. This two step opeation coesponds to adding and subtacting i (V 1 ), i.e., the position of atom i at volume V 1 in the absence of the defect, to the ight hand side of the above equation, u p0 ( i ) = i (V 1 ) i (V 1 ) + i(v 1 ) i (V 0 ). (5) What one obtains in this way is u p0 ( i ) = u V1 ( i ) + u h ( i, V 0, V 1 ), (6) whee u V1 ( i ) = i (V 1) i (V 1 ) is the displacement field obtained by inseting the defects at volume V 1 fo fixed simulation cell and u h ( i, V 0, V 1 ) = i (V 1 ) i (V 0 ) is the displacement field coesponding to a homogeneous dilatation (o contaction) of the pefect cystal without defect fom volume V 0 to volume V 1. This simple defomation coesponds to a displacement u h ( i, V 0, V 1 ) = (V 1 /V 0 ) 1/2 i (in thee dimensions the exponent is 1/3). Hence the displacement fields fo constant pessue and constant volume ae elated by: V1 u p0 ( i ) = u V1 ( i ) + i. (7) V 0 Thus, one can detemine the constant-pessue displacement at pessue p 0 by calculating the constant-volume displacement at volume V 1, the volume at pessue p 0 in the pesence of the defect. Simila consideations can be used to elate the constant-pessue and constant-volume displacement fields if the simulation cell is pemitted to change shape as well as volume duing the constant-pessue defect insetion. In this case, the simulation cell is chaacteized by the vectos a and b along its edges [33]. It then tuns out that the displacement field of a defect inseted into an initially ectangula simulation cell with edge vectos a and b at constant hydostatic pessue is simply elated to the displacement field fo fixed cell vectos a and b which, in geneal, diffe fom a and b. Fo sufficiently lage systems, howeve, a fixed shape of the simulation cell is only a vey weak constaint. In paticula, a displacement field which tends to be isotopic at lage distances may lead to a change in aspect atio of the simulation cell at constant pessue, but not to a change in the elative oientation of the edge vectos. All calculations of this pape ae caied out fo fixed and nealy squae simulation cells. 2.2 Intestitials We fist detemine the displacement field of a single intestitial paticle. This type of point defect can exist in diffeent configuations [4] with displacement fields of diffeent symmeties [34]. The thee lowest enegy stuctues ae shown in Fig. 1. In one minimum-enegy configuation, temed I 2 intestitial and shown in Fig. 1a, the exta paticle and one of the oiginal paticles aange themselves at equal distance aound the lattice position of the oiginal paticle leading to a two-fold symmety. This is the two dimensional analogue of the cowdion in an fcc cystal [35]. The displacements ae lagest on the main defect axis, which can be aligned in any of the thee low-index diections of the lattice. Since the defect symmety diffes fom that of the undelying tiangula lattice, one may wonde whethe the ectangula peiodic bounday conditions used in the calculation favo the two-fold defect symmety. Calculations caied out with hexagonal bounday conditions, howeve, yield identical esults demonstating that the defect symmety is not imposed by the symmety of the bounday conditions. Othe low-enegy defect configuations include the I 3 intestitial with thee-fold symmety shown in Fig. 1b and the I d intestitial o dumbbell intestitial shown in Fig. 1c. In the I 3 configuation, the exta paticle is located at the cente of a tiangle spanned by thee neaest neighbo lattice points and the suounding paticles ae displaced outwad with espect to thei oiginal positions. In the dumbbell configuation, the intestitial paticle and one of the oiginal paticles compete fo one lattice position as in the I 2 intestitial, but the line connecting them is othogonal to one of the low-index lattice diections. In contast to the I 2 intestitial, the I d is not concentated on one single axis. The intestitial configuations obseved in the Gaussian coe model have enegies that diffe by less than 0.1% of the total defect enegy. These enegy diffeence coespond to oughly 20% of the themal enegy k B T at melting. At finite tempeatues that ae not too low, inteconvesion between the vaious defect configuations 3

4 invesion. In paticula, the votex stuctue obseved fo the vacancy is absent in the intestitial case. In the configuation V 3 with theefold symmety, paticles patially fill the vacancy void by moving in along thee axes athe than two. On the othe thee low-index axes, paticles ae moved outwad in esponse to the emoved paticle. A vacancy configuation analogous to the I d intestitial seems not to be stable even at T = 0. A configuation pepaed in this symmety ends in an antisymmetic configuation V a (see Fig.2c). Enegetically, configuations V 2 and V a ae equal and lowe than configuation V 3 by moe than twice the themal enegy k B T at melting thus exceeding the enegy diffeence of the coesponding intestitial configuations by moe than an ode of magnitude. This enegy diffeence is less than 10% of the total defect enegy. Figue 1: Displacement fields (left hand side) and paticle configuations (ight hand side) of the I 2 intestitial (a), the I 3 intestitial (b) and the I d intestitial (c). The aows epesenting the paticle displacements ae exaggeated in length by a facto of 20 fo bette visibility. On the ight hand side, the small gey sphees indicate the sites of the pefect tiangula lattice. The blue sphees epesent paticles with 6 neighbos accoding to the Voonoi constuction (black lines). Yellow sphees ae paticles with 4 neighbos, oange and geen sphees epesent paticles with 5 and 7 neighbos, espectively. is facile and all thee of them play an impotant ole duing defect diffusion [16]. 2.3 Vacancies Also vacancies can occu in vaious configuations with displacement fields displaying quite complex pattens and symmeties lowe than that of the undelying lattice. Thee minimum enegy configuations ae shown in Fig. 2. In the the vacancy configuation V 2 (see Fig. 2a), paticles move mainly on the x-axis to patially fill the void left by a emoved paticle. As a esult, paticles above and below the void site move outwad geneating a votex-like displacement field. This two-fold vacancy V 2 has the same symmety as the I 2 intestitial, but thei displacement pattens ae not simply elated by Figue 2: Displacement fields (left hand side) and paticle configuations (ight hand side) of the V 2 vacancy (a), and the V 3 vacancy (b) and the V a vacancy (c). The aows epesenting the paticle displacements ae exaggeated in length by a facto of 20 fo bette visibility. The colo code is the same as in Fig.1. 3 Elasticity Theoy Nea the defect site non-lineaities and discete lattice effects dominate the displacement patten as evidenced 4

5 by the highly anisotopic local stuctue of vacancies and intestitials. Fa away fom the defect, howeve, the petubation of the 2d-cystal should be descibed accuately by continuum elasticity theoy. In this egime, the esponse of the system to a point defect should depend on the specific fom of the inteaction potential only though the paticula values of the elastic constants. To veify to which extent elasticity theoy is valid fo two-dimensional colloidal cystals of soft paticles, we fist eview the basic equations of elasticity theoy and then solve them fo an idealized singula defect model consisting of a pai of singula foces of equal magnitude and opposite diection [36, 37, 38]. Linea elasticity theoy is usually fomulated in tems of the symmetic stain tenso [32] ǫ ij = 1 ( ui + u ) j, (8) 2 j i whee u i and i ae the i-th component of the displacement and the position, espectively. Fo small stains, Hook s law applies and the stess σ ij is linealy elated to the stain ǫ ij, σ ij = C ijkl ǫ kl, (9) whee C ijkl is the stiffness tenso. Hee and in the following, summation ove epeated indices is implied. Fo isotopic mateials, such as two-dimensional cystals with tiangula lattice, this elation educes to σ ij = λδ ij ǫ kk + 2µǫ ij, (10) whee λ and µ ae the so-called Lamé coefficients. The Lamé coefficient µ is also called the shea modulus. In ode to calculate the displacement field geneated by a point defect one must be able to detemine how the elastic continuum eacts to extenal foces. The condition that the foces on each infinitesimal volume element balance leads to σ ij j + f i = 0, (11) whee f i is component i of a given volume foce f() acting at. Using the stess-stain elation fom Equ. (10), these equilibium conditions can be fomulated in tems of the stains athe than the stesses, λ i ǫ kk + 2µ ǫ ij j + f i = 0, (12) Inseting the definition of the stain into this equation one obtains the equilibium conditions fo the displacement field u(), (λ + µ) i u j j + µ u i + f i = 0, (13) whee = 2 / x / y 2 is the Laplace opeato. Solving this equation fo a paticula aangement of foces used to model the point defect then yields the displacement field caused by the foces. Fom the displacement field one can then detemine the enegetics of the point defect. In tems of the stain tenso and the Lamé coefficients the elastic fee enegy density of the system is given by g = λ 2 ǫ2 kk + µǫ 2 ij. (14) Accodingly, the enegy density at T = 0 is given by e = λ 2 ǫ2 kk + µǫ2 ij pǫ kk. (15) The last tem of this equation stems fom the wok done against the pessue p by the dilatation ǫ kk. The stain tenso can also be witten as the sum of a tace-fee shea and a homogeneous dilation leading to the expession ( g = µ ǫ ij 1 ) 2 2 δ ijǫ kk + K 2 ǫ2 kk, (16) whee K is the so-called bulk modulus elated to λ and µ by K = λ + µ. (17) The Poisson atio ν, i.e., the negative atio of tansvese stain to axial stain upon uniaxial loading, is given by ν = λ λ + 2µ = K µ K + µ (18) and descibes how a mateial eacts when stetched. In the next subsection we will calculate the elastic constants fo ou system at T = Elastic moduli Fo a cystal in which paticles inteact with a pai potential v() depending only on the intepaticle distance the total enegy E of N paticles is given by E = 1 v( i j ), (19) 2 i j whee i and j ae the positions of paticles i and j espectively. In this case and fo T = 0, the enegy density e 0 of the undistoted lattice, the pessue p, as well as the elastic constants K and µ can be calculated fom simple lattice sums: and e 0 = ρ v( i ), (20) 2 i p = ρ v ( i ) i, (21) 4 i K = ρ [ v ( i )i 2 8 ] v ( i ) i, (22) i i [ µ = ρ ( ) ] 2 v xi y i ( i ) + v ( i ) y4 i 2 i i 3. (23) 5

6 Hee, v () and v () ae the fist and second deivative of the pai potential, espectively, ρ is the numbe density, i is the distance of paticle i fom the oigin, and x i and y i ae its Catesian coodinates. The lattice position is chosen such that thee is one paticle at the oigin. The sums in the above equations must include a sufficient numbe of paticles to ensue convegence of the sums. (The pime on the sum symbol indicates that the paticle at the oigin is not included in the sum) a a Figue 3: Bulk modulus K (solid line), shea modulus µ (dashed line), pessue p (dash-dotted line), enegy density e (thin solid line), and Poisson atio ν (dotted line) as a function of the lattice constant a. The moduli ae given in units of ε/σ 2 and the lattice constant in units of σ. In the inset, the shea modulus µ is displayed on a lage scale. The vetical thin dotted line indicates the lattice constant a = σ coesponding to the density ρ = 0.6σ 2, at which all calculations discussed in this aticle ae caied out. The elastic constants µ, and K as well as the Poisson atio ν, the pessue p and the enegy density e, calculated using such sums, ae shown in Fig. 3 as a function of the lattice constant a, which, in a tiangula lattice, is elated to the density by ρ = 2/ 3a 2. At a density of ρ = 0.6σ 2, the density at which all calculations pesented in this aticle ae caied out, the elastic constants have the values K = εσ 2 and µ = εσ 2, the pessue is p = εσ 2, and the Poisson atio is ν = The enegy density is e 0 = εσ 2 coesponding to an enegy pe paticle of E/N = ε. Note that at this density, the system is stabilized against shea only by inteactions beyond neaest-neighbo contibutions; estimation of µ fom neaest neighbo inteactions only yields a negative shea ate at this density. While the bulk modulus K inceases monotonically with the density (and, as the pessue p, is popotional to ρ 2 fo small densities), the shea modulus eaches a maximum at a 1.67 and then apidly decays to zeo fo lattice constants lage and smalle than that. This behavio of the shea modulus is a eflection of the phenomenon of eentant melting obseved in the thee-dimensional Gaussian coe model [26, 27] and indicates that also in two dimensions a Gaussian coe cystal melts if sufficiently compessed. 3.2 Point defect model We next use elasticity theoy to detemine the displacement field ceated by intoducing an idealized point defect into a pefect isotopic cystal. The dilatation (o contaction) caused by the defect is modeled by two othogonal pais of foces. Each pai consists of two foces of equal magnitude F but opposite diections acting at two points sepaated by the distance h. If one assumes that one foce pai acts in x-diection and the othe one in y-diection and that the defect is centeed a the oigin, the total foce density is given by f() = Fδ()e x + Fδ( he x )e x Fδ()e y + Fδ( he y )e y. (24) Hee, e x and e y ae the unit vectos in x- and y- diection, espectively, and δ() is the Diac deltafunction in two dimensions. One then lets the sepaation h go to zeo and the foce F go to infinity in a way such that Fh emains constant. This defect model, in which the net foce acting on the mateial vanishes, is equivalent to inseting a small cicula inclusion into a hole of diffeent size [37]. The displacement field caused by this type of point defect can be detemined by fist calculating the Geen s function fo a singula foce and than caying out the limit h 0. Altenatively, one can cay out the limit h 0 fist and then solve the equilibium condition fo the foce density obtained in that way. In the following we will calculate the displacement field of the point defect model using this second appoach, in which peiodic bounday conditions can be taken into account paticulaly easily. Caying out the limit h 0 as descibed above the total foce density of Equ. (24) educes to f() = Fh δ(). (25) Inseting this expession into Equ. (13) one obtains the equilibium condition fo this simple point defect model, (λ + µ) i u j j + µ u i = Fh i δ(). (26) Taking the divegence on both sides yields (λ + 2µ) u j j = Fh δ(). (27) To solve this equation it suffices to find a displacement field that obeys (λ + 2µ) u j j = Fhδ(). (28) Using the Helmholtz-decomposition in two dimensions, we now wite the displacement in tems of the gadients 6

7 of two scala functions φ() and A() as a sum of an iotational and a divegence-fee pat, u i = φ i + ω ij A j, (29) whee the matix ω ij exchanges the components of the gadient and changes the sign of one of them: ω 11 = ω 22 = 0 and ω 21 = ω 12 = 1. Then, Equ. (28) becomes φ() = 2πγδ(), (30) whee we have used the fact that ω ij A/ j is divegence-fee and the paamete γ, which has the dimension of an aea, is given by γ = Fh 2π(λ + 2µ). (31) Equation (30) is the Poisson equation of electostatics fo a point chage of stength γ in two dimensions. A simila equation can be deived fo the the scala function A() by taking the 2d-voticity, defined as ω ij v j / i fo an abitay vecto field v = (v 1, v 2 ), of both sides of Equ. (26). Since the voticity of a gadient field vanishes, one obtains the bihamonic equation µ ( A()) = 0. (32) This equation is obeyed if the scala field A() is a solution of the Laplace equation A() = 0. (33) In the following, we will use the tivial solution A() = const and satisfy the bounday conditions though pope solution of the Poisson equation (30) fo the scala field φ. To do that, we note that K() = ln()/2π is a solution of K = δ() (see, fo instance, Ref. [39]), and hence we obtain the Geen s function φ() = γ ln(). (34) The coesponding displacement field u() follows by diffeentiation accoding to Equ. (34), u() = γ 2. (35) Thus, the displacement field caused by the point defect is isotopic and long-ange with a magnitude that is popotional to 1/. This esult is valid fo an infinitely extended elastic medium whee the bounday conditions u = 0 apply at infinity. This situation, howeve, does not coespond to the bounday conditions applied in compute simulations. In the following section we will discuss how to solve Equ. (30) with the appopiate bounday conditions. 3.3 Bounday conditions In compaing the esults of paticle simulations with those of elasticity theoy it is impotant to ealize that the displacement fields pedicted by continuum theoy ae of long-ange natue. Theefoe, it is cucial that coesponding bounday conditions ae used in both cases. All simulations discussed in this pape ae done with peiodic bounday conditions in ode to minimize finite size effects and peseve the tanslational invaiance of the pefect lattice. Hence, also the continuum calculations need to be caied out with peiodic bounday conditions. Fo a ectangula system with side lengths L x and L y, peiodic bounday conditions equie that u() = u(+l), whee l = (il x, jl y ) is an abitay lattice vecto with intege i and j. In the following, we will solve the Poisson equation (30) fo this type of bounday conditions. We stat by noting that the homogenous pat of the Poisson equation (30) admits the non-tivial solution φ 0 () = const that satisfies the bounday conditions. Theefoe, one needs to conside the extended Geen s function fo the solution of the geneal Poisson equation φ() = 2πρ() [39, 40]. In this case, the ight hand side of the Poisson equation must be othogonal to the solution φ 0 (), dφ 0 ()ρ() = const dρ() = 0. (36) In electostatics, this condition coesponds to chage neutality (the physical meaning of this condition in ou case will be discussed below). To satisfy this othogonality condition we must modify the Poisson equation (30) by subtacting 1/A fom the delta function, [ φ() = 2πγ δ() 1 ], (37) A whee A is the aea of the ectangula basic cell. In this modified equation, the ight hand side contains a homogeneous neutalizing backgound that exactly compensates fo the chage of the delta function. Solution of this equation yields the extended Geen s function of the poblem. To obtain a unique solution φ() of this equation one must futhemoe equie that this solution be othogonal to φ 0 (), dφ 0 ()φ() = const dφ() = 0. (38) Fo ou case this condition is ielevant, as only deivatives of φ() cay physical significance. Once the function φ() has been detemined by solving Equ. (37), the displacement field follows by diffeentiation Rigid cicula containe Befoe we embak on the solution of the extended Poisson equation (37) fo peiodic bounday conditions, we illustate the concepts intoduced above by detemining the displacement field of a point defect in an elastic 7

8 mateial enclosed in a containe with igid walls. Due to these walls, the component of the displacement field nomal to walls must vanish at the wall, u = 0. No condition applies fo the paallel component u. Fo a ectangula containe such igid wall bounday conditions ae equivalent to peiodic bounday conditions. If we assume, without loss of geneality, that the point defect is located at the cente of the ectangula peiodic cell, the component of the displacement field nomal to the bounday of the peiodic cell vanishes also in this case. In the following, we will detemine the effect of such igid bounday conditions on the displacement field of a point defect located at the cente of a cicula cavity enclosed by had walls. Fo this case, which exhibits all complications mentioned above, a simple analytical solution can be easily obtained. Conside a two-dimensional elastic isotopic mateial enclosed in a cicula containe of adius R. We choose the coodinate system such that the oigin is at the cente of the containe. To detemine the displacement field caused by a point defect of stength γ placed at the oigin we need to solve Equ. (37) unde the condition that the at distance R fom the oigin the displacement u nomal to the wall vanishes. We constuct a solution by supeposing the solution fo the extended mateial, u () = γ/ 2, and a homogeneous contaction, u c () = α, u() = u () + u c () = γ α. (39) 2 While u c () coesponds to a homogeneous contaction without shea, u () coesponds to a pue shea without dilatation (except fo = 0). To satisfy the bounday conditions at = R, we set α = γ/r 2 obtaining ( 1 u() = γ 2 1 ) R 2. (40) This displacement field coesponds to the potential ( φ() = φ () + φ c () = γ ln R R ), (41) 4 whee the last constant on the ight hand side takes cae of the condition expessed in Equ. (38). It is staightfowad to veify that φ() = φ () + φ c () = 2πγδ() 2πγ A, (42) such that the potential fom Equ. (41) satisfies the extended Poisson equation (37). A compaison of esults obtained numeically fo an I 2 intestitial and the pediction of elasticity theoy (Equ. (40)) is shown in Fig. 4. Fo the paticle system the igid containe was ealized by caying out the calculation in a lage system in which all paticles beyond a distance of R fom the oigin whee kept at fixed positions. The displacements obtained fom paticle and continuum calculations agee vey well fo all defect distances lage than about 15 lattice constants. In Fig. u 10 0 Simulation Continuum Theoy γ/ Figue 4: Displacement components u x and u y as a function of the distance (thin solid lines) fo an I 2 intestitial with its main axis oiented in x-diection. The cicula had wall containe has a adius of R = 106.8σ. Also plotted is the displacement computed fom continuum theoy accoding to Equ. (40) (dashed line) fo a defect stength of γ = σ 2 which best fits the numeical esults in the fa field and the simple 1/behavio (dotted line). 5 we depict the elative eo which we define in the following way: ξ u p() u c (). (43) u c () Hee, u p () and u c () ae the displacement fields obtained fom the paticle system and fom the pedictions of continuum theoy, espectively. The elative eo close to the defect can be lage than 100% (ed). Fo distances of > 20σ with find an eo of appoximately 1 5% (geen). Close to the igid containe the elative eo inceases again due to discete lattice effects. Simila ageement is found also fo the enegy density as shown in Fig. 6. Fo the displacement field of Equ. (40) one finds, using Equ. (15) the enegy density e() = 2µ γ2 γ2 + 2K 4 R 4 + 2p γ R 2. (44) This pediction of elasticity theoy matches the enegy density detemined numeically in y-diection (see Fig. 6). Due to the stong anisotopy of the I 2 defect, lage deviations ae obseved in x-diection. Fo distances of moe than about 15 lattice spacings the enegy density plateaus at a constant value. In this egime, the enegy density is essentially constant, e = 2pγ/R 2, and coesponds to the wok done by the defect against the pessue p. As discussed below, the plateau value of the density is elated to the neutalizing backgound on the ight hand side of Equ. (37) Ewald summation Fo an isolated defect at the cente of a ectangula cell, the symmety imposed by peiodic bounday conditions 8

9 e x-diection y-diection continuum theoy Figue 5: Colo coded elative deviation ξ calculated accoding to Equ. (43) of an I 2 intestitial at the oigin of a igid cicula box with adius R = σ. The fit paamete γ = σ 2 was found by minimizing the sum of the elative eo of paticles at distances lage than 30σ. The contou lines in white, black, and blue epesent an eo of 1%, 2%, and 5%, espectively. equies that the components of the displacement field othogonal to the suface of the cell vanish. In a fist attempt to obtain the displacement field fo such bounday conditions one may stat fom the solution fo the extended mateial and satisfy the bounday conditions by placing image defects, each of which caies the displacement field fo the infinitely extended mateial, at appopiate positions. Fo a ectangula cell, an infinite numbe of image defects aanged on a egula lattice with lattice constants L x and L y in x- and y-diection, espectively, ae equied. These image defects, which ae analogous to the image chages of electostatics, coespond to the defects in the peiodic images of the basic simulation cell. Supeposition of the displacement fields of all image defects then yields the displacement field fo peiodic bounday conditions. Due to the long-ange natue of the defect field fo the infinite mateial, howeve, such a summation of the contibution of all image defects leads to displacement fields that ae only conditionally convegent. A moe appopiate teatment that avoids this poblem consists in detemining the Geen s function of the Poisson equation (37) fo peiodic bounday conditions. The equiement imposed by the peiodic bounday conditions can be easily satisfied by expessing the solution as a Fouie seies and solving the Poisson equation in Fouie space. This teatment, howeve, leads to seies that ae only conditionally convegent with values that depend on the summation ode. The solution of this poblem using so called Ewald sums is known fom electostatics [41, 42, 40] and consists in sepaating the conditionally convegent seies into a eal space and and a Fouie Figue 6: Enegy density e as a function of distance measued along the x-axis (solid line) and the y-axis (dotted line) fo the paticle system in a cicula containe with had walls and adius R = 71.91σ. Also shown is the enegy density calculated fom continuum theoy accoding to Equ. (44) (dashed line). space pat, φ() = γ { 1 2 2π A E i [ η 2 + l 2 ] l k 0 k 2 cos(k ) + π 2η 2 A. (45) e k2 /4η 2 Hee, E i (x) = x (et /t)dt is the exponential integal and A is the aea of the ectangula cell. The fist sum is ove all lattice vectos l and the second sum is ove all ecipocal vectos k consistent with the peiodic bounday conditions. The adjustable paamete η, set to a value of η = 6/L x hee, detemines the ate of convegence of the two sums, but the value of the sums does not depend on η. The exclusion of the k = 0 tem in the above equation stems fom the equiement that both φ() and the ight hand side of the Poisson equation need to be othogonal to φ 0 () as expessed in Eqs. (36) and (38). It is easy to show by diect calculation of the Laplacian φ that the above expession fo φ() indeed obeys Equ. (37) and thus implies a neutalizing backgound of magnitude γ/a as in the pevious example. Fom Equ. (45) fo the scala function φ() the displacement field of a point defect in a system with peiodic bounday conditions is detemined by diffeentiation, { i + l i +l u i () = γ 2 + l 2 e η2 l + 2π e k2 /4η 2 A k 2 k i sin(k ). (46) k 0 Fo the systems and paametes consideed in this pape, the eal space sum can be tuncated afte the fist tem 9

10 and the Fouie space sum can be evaluated accuately using about ecipocal vectos. As mentioned above, exactly the same bounday conditions apply if the system is constained by had walls to eside in an aea of given size. Also in that case, the bounday conditions equie that the component of the displacement field othogonal to the walls vanishes. Hence, in the continuum desciption, had walls have the same effect as an infinite aay of image chages (plus neutalizing backgound ) placed on a egula lattice with a geomety detemined by the wall positions. The effect of such image chages and the neutalizing backgound is, theefoe, not a pue atifact of the peiodic bounday condition applied in the simulations, but occus also in expeimental ealizations of colloidal cystals of puely epulsive paticles which need to be kept togethe by confining walls. Accodingly, the analysis of displacement pattens (and defect inteactions) obseved expeimentally equies a simila teatment as that used hee fo the intepetation of ou simulation esults. 3.4 Electostatic analogy In electostatics, the technique of Ewald summation is used to detemine the enegetics of peiodic systems containing point chages and dipoles. When one uses this technique, one implicitly stipulates that the chages ae immesed in a homogeneous backgound that compensates fo the point chages and establishes oveall chage neutality. This neutalizing backgound is imposed by the peiodic bounday conditions; without it, no peiodic solution of the Poisson equation exists. Since mathematically the situation we face when detemining the displacement field of point defects is identical to that of electostatics, one may wonde about the physical meaning of the neutalizing backgound in ou Equ. (37). To addess this question, we note that the local volume change, o dilatation, due to a displacement field u() is given be the tace ǫ kk of the coesponding stain tenso [32]. The total change in volume V of a cetain egion G is then given as the integal ove the dilatation, V = dǫ kk (). (47) G On the othe hand, it follows fom the definition of φ (see Equ. (29)) that the tace of the stain tenso is equal to the Laplacian of φ, φ() = ǫ kk (). (48) Thus, the Poisson equation (37) is an equation fo the local dilatation. Accoding to this equation, at the defect site the dilatation is equied to have a delta like peak of stength 2πγ. The total volume change caused by this singula dilatation, V = d2πγδ() = 2πγ, A is exactly compensated by the total volume change oiginating fom the constant neutalizing backgound, V = d2πγ/a = 2πγ. (Calculating the stain A tenso diectly fom the displacement field of Equ. (46) indeed yields ǫ kk = 2πγ/A.) Theefoe, the condition of chage neutality of electostatics coesponds to the equiement of constant volume in ou case. In this analogy, the chage density of electostatics coesponds to the local dilatation (the chage coesponds to the volume change) and the ole of the electic field is played hee by the displacement field. This intepetation of the Poisson equation (37) also suggests a definition of the defect volume V d. As mentioned above, intoduction of a point defect of stength γ leads to a total volume change which can be viewed as the volume of the defect, V d = 2πγ. (49) With peiodic (o igid) bounday conditions the system as a whole is pevented fom changing volume and the volume change due to the defect is exactly compensated by the homogeneous neutalizing backgound. This defect volume is also what one gets when calculating the expansion of a cicle unde the 1/-defomation caused by the idealized defect model (see Equ. (35)). Measuing the paamete γ, fo instance by fitting the displacement field fa fom the defect to the continuum theoy esults, thus pemits to detemine the defect volume. Fo the I 2 at a density of ρ = 0.6σ 2, fo example, we found a volume of V d = 1.44σ 2, which is slightly smalle then V 0 = 1.66σ 2, the volume pe paticle in the pefect lattice. The neutalizing backgound appeaing in the Poisson equation (37) also figues in the enegy density and can contibute consideably to the total defect enegy. Accoding to Equ. (15), the enegy density includes the tem e p = ǫ kk p aising fom the wok caied out by the defect against the pessue p. Away fom the singulaity at the oigin, this component of the enegy density is constant, e p = 2πγ/A, and it dominates fo lage distances fom the defect. Although e p is popotional to 1/A and theefoe small in geneal, integating it ove the entie aea (leaving out the unphysical singulaity at the oigin) yields an enegy contibution of E p = 2πγ which is independent of system size and can be substantial. Fo an I 2 intestitial in ou Gaussian coe system at ρ = 0.6σ 2, fo instance, this contibution amounts to moe than 50% of the total defect enegy. Inteestingly, no such pessue contibution aises fo the displacement field of Equ. (35) obtained fo the infinitely extended mateial. Thus, the condition of fixed volume imposed by the peiodic bounday conditions leads to measuable effects also in the lage system limit and even bounday conditions applied at infinity matte. 4 Compaison of simulation and continuum theoy In this section we compae the esults of the paticlebased simulations with the pedictions of continuum elasticity theoy obtained in the pevious section. In paticula, we veify at which distances fom the point defect elasticity theoy becomes valid and which effect 10

11 bounday conditions have on the displacement fields. We fist conside the displacement fields of intestitials, then those of vacancies. 4.1 Intestitials As discussed in Sec. 2, insetion of an intestitial paticle into a pefect lattice can lead to diffeent displacement pattens, all of which ae highly anisotopic nea the defect site. Fathe away fom the defect the anisotopy should subside as the isotopic behavio expected fom elasticity theoy sets in. This is indeed what is obseved fo intemediate distances fom the defect as shown in Fig. 7 fo an I 2 defect. In the bottom panel of this figue, the displacement magnitude u() is plotted as a function of the distance fom the defect. Each dot coesponds to one paticula paticle. While fo shot distances the displacement magnitude is not a unique function of due to the anisotopy of the defect, at lage distances u() is essentially detemined by. In this intemediate egime, the displacement magnitude seems to follow the 1/-behavio pedicted by elasticity theoy fo the infinitely extended mateial. Due to the peiodic bounday conditions, howeve, the displacement magnitude cannot emain isotopic as the bounday is appoached. In fact, the peiodic bounday conditions lead to a spead of u() at lage distances. The two pongs obseved in the bottom panel of Fig. 7 coespond to the diections along the x- and y-axes and along the diagonals. θ() u() π/2 π/ Figue 7: Top: Angle θ between the displacement vecto u and the position vecto as a function of the distance fom the defect fo an I 2 intestitial in a paticle system. Each dot coesponds to one paticle. Bottom: Displacement magnitude u() as a function of. Also shown as a dashed line is the γ/-line fo γ = σ 2. This kind of behavio is obseved even moe clealy fo the displacement diections. In the top panel of Fig. 7, the angle θ between the displacement u() and the position vecto is plotted as a function of distance. As in the bottom panel, each dot coesponds to an individual paticle. Fo an isotopic displacement field, the displacement and the position vecto ae pefectly aligned and θ = 0. Thus, non-zeo angles θ ae an indication of anisotopy. Fo small distances angles θ lage than π/4 occu. Fo intemediate distances, 20 < < 150, the angle θ is small since in this egime the displacements appoximately points away fom the oigin. Fo lage distances, the peiodic bounday conditions then lead to a spead in θ and deviations of up to θ = π/2 ae possible. u u Simulation Ewald summation exp(-α) γ/ Figue 8: Displacement components u x and u y of the I 2 intestitial as a function of distance along the x- axis and y-axis, espectively (solid lines). The main axis of the I 2 defect is oiented in x-diection. Also plotted ae the displacement computed fom continuum theoy by Ewald summation accoding to Equ. (46) (dashed line), and simple 1/-behavio (dotted line). Fo shot distances, the behavio of the displacement along the x-axis is exponential (dash-dotted line) as descibed by a simple bead-sping model [34]. The inset shows the egion close to the defect location. A defect stength of γ = σ 2 was used hee since this value yields the best fit of the esults obtained fom elasticity theoy and the numeical esults at lage distances fom the defect. The long-distance behavio descibed above is pefectly epoduced by linea elasticity theoy. As shown in Fig. 8 fo I 2 intestitials and in Fig. 9 fo I 3 and I d intestitials, espectively, the displacement calculated using Ewald summation accoding to Equ. (46) agees vey well with the numeical esults fo all distances lage than about lattice spacings. In paticula, the deviations fom the 1/-behavio nea the cell bounday ae pefectly captued by elasticity theoy with peiodic bounday conditions. Fo small distances, on the othe hand, the displacement field in the paticle system is highly anisotopic with stong deviations between the x- and y-diection. In this non-linea coe egion elasticity theoy is not applicable and the displacements of the thee configuations diffe. The ex- 11

12 u 10 0 u u x 10 0 u x Simulation Ewald summation γ/ u u Simulation Ewald Summation γ/ Figue 9: Top: Displacement components u x and u y of the I 3 intestitial as a function of distance along the x-axis and y-axis, espectively (solid lines). Also plotted ae the displacements computed fom continuum theoy by Ewald summation accoding to Equ. (46) (dashed line), and simple 1/-behavio (dotted line). The inset shows the egion close to the defect location. A defect stength of γ = σ 2 was used. Bottom: Displacement components as above fo an I d intestitial with γ = σ 2. ponential shot-ange dependence of u x on the distance fo I 2 intestitials is, howeve, captued by a simple bead-sping model discussed in Ref. [34]. In this model, the exponential decay constant can be elated to the elastic constants of the mateial. In the compaison of the esults obtained fo the paticle system with those of continuum theoy the defect stength γ is teated as an adjustable paamete. Fo each configuation, I 2, I 3, and I d, the paticula displacement stength γ was found by optimizing the elative eo (see Equ. (43)) at distances lage than 30.0σ fom the oigin of the defect. Fo the I 2 defect, a value of γ = σ 2 yields the best fit. I 3 and I d intestitials poduce a slightly lage displacement with a stength of γ = σ 2 and γ = σ 2, espectively. The question aises if this fit is independent on the size of the box. In Fig. 10 we depict the displacement of an I 2 defect fo diffeent box sizes togethe with the esults fom the Ewald summation. The displacements plotted in the inset of Fig. 10 on a doubly-logaithmic scale clealy in Figue 10: Displacement component u x as a function of distance fom the defect along the x-axis obtained fom simulations (solid lines) and accoding to Equ. (46) (dashed lines) fo system sizes N = 26 30, 52 60, 78 90, , , , , Also shown is the γ/ behavio expected in an infinitely extended mateial (dotted line). The same value of γ = σ 2 was used in all cases. The vetical dotted lines indicate the distances of the cell boundaies fom the oigin fo the vaious system sizes. In the inset the same cuves ae displayed on a logaithmic scale. dicate that the algebaic 1/ behavio is obseved, if at all, only fo lage system sizes and in a limited distance ange. A compaison of the Ewald summation esults with the numeical calculations ove the whole simulation cell is shown in Fig. 11. The colo coded map epesents the elative deviation (see Equ. (43)) of u p () fom u c (). In the figue, egions of lage and small elative deviation ae coloed in ed and blue, espectively. We find that the I 2 and I 3 configuation show a elative deviation between 1% to 5% ove the whole ange. Only in the coe egion of the defect the deviations ae lage. Fo the I d defect the deviations ae lage and between 10% to 20% also fa away fom the defect. These deviations ae due to discepancies both in oientations as well as magnitude. The enegy density of a point defect with peiodic boundaies, calculated fom Equ. (15) fo the displacement field of Equ. (46), is depicted in Fig. 12. As fo the point defect in a cicula igid containe discussed in Sec , the enegy density becomes constant fo lage distances. This constant tem aises fom the wok pefomed by the defect against the extenal pessue. 4.2 Vacancies In the system studied in this pape, vacancies geneate displacement pattens that ae consideably moe inticate than those of intestitials, as can be infeed fom a compaison of Figs. 1 and 2. While in the case of intestitials the displacement vectos essentially point away fom the defect site, vacancies have displacement fields which point outwad o inwad depending 12

13 on the position elative to the defect. Fo instance, in the V 2 vacancy shown in Fig. 2a the displacement vectos point towads the defect site along the x-axis, but away fom the defect along the y-axis. Between the two axes, votex like stuctues occu. A simila displacement patten with altenating displacement diections foms also fo the V a vacancy shown in Fig. 2c. This behavio obseved in the coe egion aound the defect can not be epoduced by the simple defect model used hee fo the continuum theoy calculations. The displacement field obtained in this model is eithe oiented towads the defect o away fom it depending on the sign of the defect stength γ. Fo the V 3 vacancy, on the othe hand, all displacement vectos point inwad and no such complications occu. Nevetheless, the displacement magnitudes shown in Fig. 13 fo the thee vacancy configuations follow qualitatively the fom pedicted by continuum theoy. Fo all thee configuations, vaying the defect stength γ can lead to bette ageement in paticula diections (e.g., along the x- o y-axis), but not ove the entie plane. In Fig. 14 we depict the elative deviations of the displacement of the paticle simulation fom the continuum theoy calculated accoding to Equ. 43. Even fa fom the defect the displacement field obtained fom the paticle simulation does not become isotopic such that it cannot be epoduced by the continuum theoy calculations. The best ageement is found fo V 3 (Fig. 14, cente), in which case the anisotopy of the displacement field is less ponounced. The failue of the point defect model to epoduce the displacement pattens of vacancies, howeve, does not imply that elasticity theoy is unsuitable fo the desciption of such defects. Rathe, the point defect model used hee, which consists of two othogonal pais of opposing foces, appeas to be to simple to captue the complex displacement pattens induced by vacancies. 5 Conclusion Intestitials and vacancies occu in vaious configuations geneating displacement fields with symmeties that diffe fom the symmety of the undelying tiangula lattice. Nea the defect, the displacement fields ae highly anisotopic and stongly dependent on the atomistic details of the inteactions. In this distance egime, linea elasticity theoy bakes down due to discete lattice effects and non-lineaities of the potential. Fo distances lage than about lattice spacings, howeve, elasticity theoy is valid. To establish this validity, it is cucial that coesponding bounday conditions ae used both in the continuum calculations and the paticlebased numeical simulations. Since simulations ae usually caied out with peiodic bounday conditions in ode to minimize finite size effects, the same bounday conditions must be employed also in the continuum calculation. If diffeent bounday conditions ae used, the long-ange natue of elastic displacement fields can lead to consideable discepancies even at length scale whee elasticity theoy is expected to hold. In this pape we have fomulated the elastic theoy poblem in a way that makes it fomally identical to the poblem of detemining the potential of a point chage in electostatics. While hee we have focused on two-dimensional systems, the same fomalism applies also to thee dimensions. Unde peiodic bounday conditions, this two-dimensional electostatics poblem has been solved using the method of Ewald summation [41, 42], in which the solution is expessed in tems of two apidly convegent sums, one in eal space and one in ecipocal space. The solution of the electostatics poblem can be simply tansfeed to the continuum theoy of the point defect. In this case, the ole of the chage density in electostatics is played by the dilatation, i.e. the local elative volume change. Accodingly, the chage neutality equied by the peiodic bounday conditions in electostatics coesponds to the condition of fixed volume in the elasticity theoy. This equiement leads to a homogeneous neutalizing backgound that is seamlessly incopoated in the Ewald sum solution. The neutalizing backgound satisfies the condition of fixed volume by exactly compensating fo the volume change caused by the intoduction of the point defect. The volume compensation leads to an additional tem in the enegy density elated to the wok done by the defect against the extenal pessue. Depending on the pessue, this enegy can contibute significantly to the total defect enegy. While fo intestitials the elasticity theoy calculations caied out fo a simple point defect model lead to good ageement with the paticle calculations in the coe egion aound the defect, lage deviations ae obseved fo vacancies. These discepancies ae due to the moe complex displacement pattens of vacancies and bette defect models ae equied to captue this behavio. Acknowledgments The authos would like to thank Chistos Likos, Matin Neumann, and Andeas Töste and fo useful discussions. This eseach was suppoted by the Univesity of Vienna though the Univesity Focus Reseach Aea Mateials Science (poject Multi-scale Simulations of Mateials Popeties and Pocesses in Mateials ). Refeences [1] U. Gasse, E. R. Weeks, A. Schofield, P. N. Pusey, and D. A. Weitz, Science 292, 258 (2001). [2] V. Pasad, D. Semwogeee, and E. R. Weeks, J. Phys.: Cond. Mat. 19, (2007). [3] A. Petsinidis and X. S. Ling, Natue 413, 147 (2001). 13

14 [4] A. Petsinidis and X. S. Ling, New J. Phys. 7, 33 (2005). [5] A. Petsinidis and X. S. Ling, Phys. Rev. Lett. 87, (2001). [6] K. Zahn, R. Lenke, and G. Maet, Phys. Rev. Lett. 82, 2721 (1999). [7] C. Eisenmann, U. Gasse, P. Keim, G. Maet, H.-H. von Günbeg, Phys. Rev. Lett. 95, (2005). [8] C. Eisenmann, U. Gasse, P. Keim, G. Maet, and H.H. von Günbeg, Phys. Rev. Lett. 95, (2005). [9] J. Kostelitz and D. Thouless, J. Phys. C 6, 1181 (1973); B. Halpein and D. Nelson, Phys. Rev. Lett. 41, 121 (1978). [10] D. S. Fishe, B. I. Halpein, and R. Mof, Phys. Rev. B 20, 4692 (1979). [11] K. Bagchi, H. C. Andesen, and W. Swope, Phys. Rev. E 53, 3794 (1995). [12] K. Bagchi, H. C. Andesen, and W. Swope, Phys. Rev. Lett. 76, 255 (1995). [13] H. C. Andesen and W. Swope, J. Chem. Phys. 102, 2851 (1995). [14] E. Kim and M. H. W. Chan, Natue 427, 225 (2004). [15] P. N. Ma, L. Pollet, M. Toye, and F. C. Zhang, axiv:cond-mat, v2 (2007). [16] A. Libal, C. Reichhadt, and C.J. Olson Reichhadt, Phys. Rev. E, (2007). [17] L. C. DaSilva, L. Candido, L. D. F. Costa, O. N. Oliveia, Phys. Rev. B 76, (2007). [18] W. Cai, V. V. Bulatov, J. Chang, J. Li, and S. Yip, Phys. Rev. Lett. 86, 5727 (2001). [19] W. Cai, V. V. Bulatov, J. Chang, J. Li, and S. Yip, Phil. Mag. 83, 539 (2003). [20] D. Fenkel and B. Smit, Undestanding Molecula Simulation, Academic Pess, San Diego (2002). [21] A. J. C. Ladd and W. G. Hoove, Phys. Rev. B 26, 5469 (1982). [25] C. R. Nugent, K. V. Edmond, H. N. Patel, E. R. Weeks, Phys. Rev. Lett. 99, (2007). [26] F. H. Stillinge, J. Chem. Phys. 65, 3968 (1976). [27] S. Pestipino, F. Saijta and P. V. Giaquinta, Phys. Rev. E 71, (R) (2005). [28] S. Pestipino, F. Saijta and P. V. Giaquinta, J. Chem. Phys. 123, (2005). [29] P. J. Floy, Pinciples of Polyme Chemisty, Conell Univesity Pess, Ithaca (1953). [30] F. H. Stillinge and T. A. Webe, J. Chem. Phys. 74, 4015 (1981). [31] F. H. Stillinge and T. A. Webe, J. Chem. Phys. 74, 4020 (1981). [32] L. Landau and E. Lifschitz, Theoy of Elasticity, Pegamon, London (1959). [33] M. Painello and A. Rahman, Phys. Rev. Lett. 45, 1196 (1980). [34] W. Lechne, E. Schöll-Paschinge, C. Dellago, J. Phys.: Cond. Mat., submitted (2008). [35] L. Tewodt, Phys. Rev. 109, 1 (1958). [36] J. D. Eshelby, Phil. Tans. R. Soc. A, 244, 87 (1951). [37] J. D. Eshelby, Act. Met. 3, 487 (1955). [38] D. J. Bacon, D. M. Banett, and R. O Scattegood, Pog. Mat. Sci. 23, 51 (1979). [39] R. Couant and D. Hilbet, Methods of Mathematical Physics I, Wiley, New Yok (1953). [40] M. Neumann, pesonal communication (2008). [41] S.W. de Leeuw and J.W. Peam, Physica 113A, (1982). [42] R. Kach, M. Neumann, F. Neumann, R. Ullich, J. Neumülle, and W. Scheine, Physica 369A, (2006). [43] J. P. Hith and J. Lothe, Theoy of Dislocations, Kiege Publishng Company, Malaba, Floida (1992). [22] P. B. Bladon and D. Fenkel, J. Phys. Chem. 180, 6707 (2004). [23] A. D. Dinsmoe, E. R. Weeks, V. Pasad, A. C. Levitt, and D. A. Weitz, Appl. Opt. 40, 4152 (2001). [24] D. G. A. L. Aats, R. P. A. Dullens, and H. N. W. Lekkekeke, New J. Physics 7, 40 (2005). 14

15 e x-diection y-diection continuum theoy Figue 12: Enegy density e of an I 2 intestitial in a peiodic box as a function of the distance measued along the x-diection (solid line) and the y-diection (dotted line). Also shown is the enegy density obtained fom continuum theoy (dashed line) accoding to Equ. (15) fo the displacement of Equ. (46). Figue 11: Colo coded elative deviations of the displacement u p () obtained numeically fom the continuum theoy pediction u c () as a function of x and y calculated accoding to Equ. (43). Fom top to bottom, the figues depict the elative deviations fo the I 2 intestitial (γ = σ 2 ), the I 3 intestitial (γ = σ 2 ), and the I d intestitial (γ = σ 2 ). The whole simulation cell of dimensions L x = σ and L y = σ is shown. Colos ae assigned on a logaithmic scale which uns fom 10 3 (blue) to 10 1 (ed). The white, black, and blue contou lines epesent a elative eo of 1%, 2% and 5%, espectively. 15

16 10 0 u u Simulation x-diection Simulation y-diection Ewald Summation γ/ Simulation x-diection Simulation y-diection Ewald Summation γ/ u Simulation x-diection Simulation y-diection Ewald Summation γ/ Figue 13: Fom top to bottom: Absolute value of the displacement components u x (black solid line) and u y (black dashed line) of a V 2, V 3 and V a vacancy. Also plotted ae the absolute values of the pedicted displacement field fom the Ewald summation Equ. (46) fo the defect stengths γ = 1.78σ 2 (V 2 ), γ = 0.294σ 2 (V 3 ), and γ = 1.89σ 2 (V a ), found by minimizing the elative eo of Equ. 43 ove the whole xy-plane (dashed line), as well as 1/ behavio (dotted line). Figue 14: Colo coded elative deviations Equ. (43) of the displacement obtained numeically fo thee diffeent vacancy configuations fom the pediction of continuum theoy as a function of x and y. Colo code and system size is the same as in Fig. (11). Fom top to bottom: V 2 vacancy (γ = 1.78σ 2 ), V 3 vacancy (γ = 0.294σ 2 ), and V a vacancy (γ = 1.89σ 2 ). 16